arXiv subjects

Jonathan Wittmer

Publications and source records attributed to Jonathan Wittmer.

5 recordsLinked to original sources

arXiv2026

A New Look at the Ensemble Kalman Filter for Inverse Problems: Duality, Non-Asymptotic Analysis and Convergence Acceleration

This work presents new results and understanding of the Ensemble Kalman filter (EnKF) for inverse problems. In particular, using a Lagrangian dual perspective we show that EnKF can be derived from the sample average approximation (SAA) of the Lagrangian dual function. The beauty of this new duality perspective is that it facilitates us to prove and numerically verify a novel non-asymptotic convergence result for the EnKF. Motivated by the new perspective, we also present a new convergence improvement strategy for the Ensemble Kalman Inversion Algorithm (EnKI), which is an iterative version of the EnKF for inverse problems. In particular, we propose an adaptive multiplicative correction to the sample covariance matrix at each iteration and we call this new algorithm as EnKI-MC (I). Based on the new duality perspective, we derive an expression for the optimal correction factor at each iteration of the EnKI algorithm to accelerate the convergence. In addition, we also consider an ensemble specific multiplicative covariance correction strategy (EnKI-MC (II)) where a different correction is employed for each ensemble. By viewing EnKI through the lens of fixed-point iteration, we also provide theoretical results that guarantees the convergence of EnKI-MC (I) algorithm. Numerical investigations for the deconvolution problem, initial condition inversion in advection-convection problem, initial condition inversion in a Lorenz 96 model, and inverse problem constrained by elliptic partial differential equation are conducted to verify the non-asymptotic results for EnKF and to assess the performance of convergence improvement strategies for EnKI. The numerical results suggest that the proposed strategies for EnKI not only led to faster convergence in comparison to the currently employed techniques but also better quality solutions at termination of the algorithm.

math.NA↗

arXiv2025

Aria Gen 2 Pilot Dataset

The Aria Gen 2 Pilot Dataset (A2PD) is an egocentric multimodal open dataset captured using the state-of-the-art Aria Gen 2 glasses. To facilitate timely access, A2PD is released incrementally with ongoing dataset enhancements. The initial release features Dia'ane, our primary subject, who records her daily activities alongside friends, each equipped with Aria Gen 2 glasses. It encompasses five primary scenarios: cleaning, cooking, eating, playing, and outdoor walking. In each of the scenarios, we provide comprehensive raw sensor data and output data from various machine perception algorithms. These data illustrate the device's ability to perceive the wearer, the surrounding environment, and interactions between the wearer and the environment, while maintaining robust performance across diverse users and conditions. The A2PD is publicly available at projectaria.com, with open-source tools and usage examples provided in Project Aria Tools.

cs.CV↗

arXiv2023

An autoencoder compression approach for accelerating large-scale inverse problems

PDE-constrained inverse problems are some of the most challenging and computationally demanding problems in computational science today. Fine meshes that are required to accurately compute the PDE solution introduce an enormous number of parameters and require large scale computing resources such as more processors and more memory to solve such systems in a reasonable time. For inverse problems constrained by time dependent PDEs, the adjoint method that is often employed to efficiently compute gradients and higher order derivatives requires solving a time-reversed, so-called adjoint PDE that depends on the forward PDE solution at each timestep. This necessitates the storage of a high dimensional forward solution vector at every timestep. Such a procedure quickly exhausts the available memory resources. Several approaches that trade additional computation for reduced memory footprint have been proposed to mitigate the memory bottleneck, including checkpointing and compression strategies. In this work, we propose a close-to-ideal scalable compression approach using autoencoders to eliminate the need for checkpointing and substantial memory storage, thereby reducing both the time-to-solution and memory requirements. We compare our approach with checkpointing and an off-the-shelf compression approach on an earth-scale ill-posed seismic inverse problem. The results verify the expected close-to-ideal speedup for both the gradient and Hessian-vector product using the proposed autoencoder compression approach. To highlight the usefulness of the proposed approach, we combine the autoencoder compression with the data-informed active subspace (DIAS) prior to show how the DIAS method can be affordably extended to large scale problems without the need of checkpointing and large memory.

math.NA↗

arXiv2023

On Unifying Randomized Methods For Inverse Problems

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation. More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as a sample average approximation enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. We present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.

math.NA↗

arXiv2021

Solving Bayesian Inverse Problems via Variational Autoencoders

In recent years, the field of machine learning has made phenomenal progress in the pursuit of simulating real-world data generation processes. One notable example of such success is the variational autoencoder (VAE). In this work, with a small shift in perspective, we leverage and adapt VAEs for a different purpose: uncertainty quantification in scientific inverse problems. We introduce UQ-VAE: a flexible, adaptive, hybrid data/model-informed framework for training neural networks capable of rapid modelling of the posterior distribution representing the unknown parameter of interest. Specifically, from divergence-based variational inference, our framework is derived such that most of the information usually present in scientific inverse problems is fully utilized in the training procedure. Additionally, this framework includes an adjustable hyperparameter that allows selection of the notion of distance between the posterior model and the target distribution. This introduces more flexibility in controlling how optimization directs the learning of the posterior model. Further, this framework possesses an inherent adaptive optimization property that emerges through the learning of the posterior uncertainty.

stat.ML↗